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MEN  OF MATHEMATICS

sis is. It occurs in the famous memoir Ueber die Anzahl der
Primzahlen unter einer gegebenen Grosse (On the number of
prime numbers under a given magnitude), printed in the
monthly notices of the Berlin Academy for November 1859,
when Riemann was thirty-three. The problem concerned is to
give a formula which will state how many primes there are less
than any given number n. In attempting to solve this Riemann
-was driven to an investigation of the infinite series

in which s is a complex number, say s = u + iv (i = V  1),
where u and v are real numbers, so chosen that the series con-
verges, With this proviso the infinite series is a definite function
of s, say  (s) (the Greek zeta, , is always used to denote this
function, which is called 'Riemann' s zeta function'); and as s
varies,  (s) continuously takes on different values. For what
values of $ will  (s) be zero? Riemann conjectured that all such
values of s for which u lies between 0 and 1 are of the form  -f
iv, namely, all have their real part eqiial to J.
This is the famous hypothesis. Whoever proves or disproves
it will cover himself with glory and incidentally dispose of many
extremely difficult questions in the theory of prime numbers,
other parts of the higher arithmetic, and in some fields of ana-
lysis. Expert opinion favours the truth of the hypothesis. In
1914 the English mathematician G. H. Hardy proved that an
infinity of values of s satisfy the hypothesis, but an infinity is
not necessarily all. A decision one way or the other disposing of
Riemann's conjecture would probably be of greater interest to
mathematicians than a proof or disproof of Fermat's Last
Theorem. Riemann's hypothesis is not the sort of problem that
can be attacked by elementary methods. It has already given
rise to an extensive and thorny literature.
Legendre was not the only great mathematician whose
works Riemann absorbed by himself - always with amazing
speed - at the Gymnasium; he became familiar with the cal-
culus and its ramifications through the study of Euler. It is
rather surprising that from such an antiquated start in analysis
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